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Hometotally bounded uniform space

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# totally bounded uniform space

A uniform space $X$ with uniformity $\mathcal{U}$ is called *totally bounded* if for every entourage $U\in\mathcal{U}$, there is a finite cover $C_{1},\ldots,C_{n}$ of $X$, such that $C_{i}\times C_{i}\in U$ for every $i=1,\ldots,n$. $\mathcal{U}$ is called a totally bounded uniformity.

Remark. A uniform space is compact (under the uniform topology) iff it is complete and totally bounded.

# References

- 1
S. Willard,
*General Topology*, Addison-Wesley, Publishing Company, 1970.

Defines:

totally bounded, totally bounded uniformity

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## Mathematics Subject Classification

54E35*no label found*

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## Comments

## An alternate definition of totally bounded uniform space

## An alternate definition of totally bounded uniform space 2

Sorry, my wrong LaTeX. I repeat the comment again.

Minimus Heximus user of math.stackexchange.com has given in his answer an other definition:

$(X,\mathcal{D})$ is totally bounded

^{}, when for each entourage $D\in\mathcal{D}$, there are $x_{1},...,x_{n}\in X$ with $D[x_{1}]\cup...\cup D[x_{n}]=X$.Minimus Heximus has proved that his definition follows from PlanetMath definition.

Does the converse hold? Is the PlanetMath’s definition a consequence of Minimus Heximus’s definition? Or is there a counter-example?